A problem of geometric Ramsey theory is to find an isometric copy of all large dilates of a given finite point configuration in sets of positive upper density. A result of this type was obtained by Katznelson and Weiss, namely that sets of positive upper density of R^d contain all large distances. This was generalised by Bourgain to show that such sets contain isometric copies of all large dilates of a given simplex. We extend these results to show that above phenomena holds for k-dimensional rectangles and more generally for direct products of simplices.

The main tool is an adaptation to of the so-called regularity lemma (and its extension to hypergraphs) to the continuous settings with respect to certain norms controlling such configurations.